YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRS Reverse [EQUIVALENT, 0 ms]
(2) QTRS
(3) Strip Symbols Proof [SOUND, 0 ms]
(4) QTRS
(5) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms]
(6) YES


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(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a(b(a(a(b(a(a(a(a(a(x1)))))))))) -> a(a(a(a(a(a(b(a(a(b(a(a(b(x1)))))))))))))

Q is empty.

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(1) QTRS Reverse (EQUIVALENT)
We applied the QTRS Reverse Processor [REVERSE].
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(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a(a(a(a(a(b(a(a(b(a(x1)))))))))) -> b(a(a(b(a(a(b(a(a(a(a(a(a(x1)))))))))))))

Q is empty.

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(3) Strip Symbols Proof (SOUND)
We were given the following TRS:

   a(a(a(a(a(b(a(a(b(a(x1)))))))))) -> b(a(a(b(a(a(b(a(a(a(a(a(a(x1)))))))))))))

By stripping symbols from the only rule of the system, we obtained the following TRS [ENDRULLIS]: 

   a(a(a(a(a(b(a(a(b(x))))))))) -> b(a(a(b(a(a(b(a(a(a(a(a(x))))))))))))

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(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a(a(a(a(a(b(a(a(b(x))))))))) -> b(a(a(b(a(a(b(a(a(a(a(a(x))))))))))))

Q is empty.

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(5) RFCMatchBoundsTRSProof (EQUIVALENT)
Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. This implies Q-termination of R.
The following rules were used to construct the certificate:

   a(a(a(a(a(b(a(a(b(x))))))))) -> b(a(a(b(a(a(b(a(a(a(a(a(x))))))))))))

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
566, 567, 568, 569, 570, 571, 572, 573, 574, 575, 576, 577, 578, 579, 580, 581, 582, 583, 584, 585, 586, 587, 588, 589

Node 566 is start node and node 567 is final node.

Those nodes are connected through the following edges:

* 566 to 568 labelled b_1(0)* 567 to 567 labelled #_1(0)* 568 to 569 labelled a_1(0)* 569 to 570 labelled a_1(0)* 570 to 571 labelled b_1(0)* 571 to 572 labelled a_1(0)* 572 to 573 labelled a_1(0)* 573 to 574 labelled b_1(0)* 574 to 575 labelled a_1(0)* 574 to 579 labelled b_1(1)* 575 to 576 labelled a_1(0)* 575 to 579 labelled b_1(1)* 576 to 577 labelled a_1(0)* 576 to 579 labelled b_1(1)* 577 to 578 labelled a_1(0)* 577 to 579 labelled b_1(1)* 578 to 567 labelled a_1(0)* 578 to 579 labelled b_1(1)* 579 to 580 labelled a_1(1)* 580 to 581 labelled a_1(1)* 581 to 582 labelled b_1(1)* 582 to 583 labelled a_1(1)* 583 to 584 labelled a_1(1)* 584 to 585 labelled b_1(1)* 585 to 586 labelled a_1(1)* 585 to 579 labelled b_1(1)* 586 to 587 labelled a_1(1)* 586 to 579 labelled b_1(1)* 587 to 588 labelled a_1(1)* 587 to 579 labelled b_1(1)* 588 to 589 labelled a_1(1)* 588 to 579 labelled b_1(1)* 589 to 567 labelled a_1(1)* 589 to 579 labelled b_1(1)


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(6)
YES